An equity options trading desk of a financial institution frequently buys and sells options for customers and for proprietary trading. An option C is a contract between two parties that gives the buyer the right, but not the obligation, to purchase or sell a specified asset S (called the underlying asset) at a later date at an agreed upon price K (called the strike price). An option to buy the asset is referred to as a call, an option to sell the asset is referred to as a put. The purchase and sale of options involves several types of risk for the investor. One of the risks associated with options is directional risk. Directional risk is a risk that the value of an option or a portfolio of options will be negatively affected by changes in the value of the underlying asset. Financial institutions may reduce directional risk through a number of techniques, one of which is delta-hedging of the option position. Delta hedging involves buying and selling the underlying asset based on the rate of change in the price of an option with respect to the price of the underlying asset (i.e., based on the delta (Δ) of the instrument). Delta is defined as the first partial derivative of the option price C with respect to the stock price S. In mathematical terms, delta is expressed as follows:
      Δ    =                  ∂        C                    ∂        S              ,
Assuming that a delta value is constant, if the price of a particular stock changes by an amount (X), then an option on that stock will change by (Δ) times (X). Thus, if the delta of a call option is 0.4, and the stock price changes by $1 then the option price is expected to change by about 40% of that amount ($0.40). In this case, if an investor holds one call option and sells short Δ of the underlying stock (a “hedge position” on the investment), the investor will be immunized against changes in the option price due to small movements of the stock price.
In practice, the delta of an option is not constant. Thus, as the stock price changes, the delta likewise changes. When the change in the stock price becomes fairly large, the delta itself will change significantly. To maintain immunization against changes in the option price, the investor will need to adjust the hedge position. If, for example, the trader established an initial hedge position at an initial delta value of Δ0 and the changed delta value becomes Δ1, then the trader will need to adjust the hedge position by the difference (Δ0−Δ1) between the original and new delta values. If the difference is positive, the investor will need to purchase (Δ0−Δ1) shares of stock for each option position; if the difference is negative, the investor will need to sell (Δ0−Δ1) shares. This whole process of frequent adjustments is called delta-hedging and continues until option maturity or until the trader sells the option back to the market.
As a practical matter, to apply delta hedging, a trader must be able to approximate the change in delta as the price of a stock changes. The change in delta may be approximated based on the Gamma (Γ) of the option. Gamma is a measure of the rate of change of delta with respect to changes in the price of the underlying asset. Gamma is defined as the first partial derivative of delta with respect to the stock price S. By extension, Γ is equivalently defined as the second partial derivative of the option price C with respect to the stock price S. In mathematical terms, gamma is expressed as follows:
      Γ    =                            ∂          Δ                          ∂          S                    =                        ∂                      C            2                                                ∂            2                    ⁢          S                      ,For a change in stock price of (S0−S1), where S0 is the initial stock price and S1 is the new stock price, and assuming that the gamma (Γ) of an option tends to remain relatively constant with respect to changes in stock price during a trading day, the change in value of the delta (which determines the quantity of stock to be purchased or sold to maintain a hedge position) can be approximated as follow:Δ0−Δ1≅Γ×(S0−S1)
If there is an unusually large change in stock price, or if the option is close to maturity with S close to K, the assumption of a constant gamma may be inappropriate and intra-day adjustments to the gamma value may be required.
If a trader has bought a plain call or put option, then the gamma of the option position is positive (commonly referred to as “long gamma”). FIG. 1 shows a simple long gamma position for a plain vanilla European Call option with a strike price K=100, a time to maturity T=1 year, on a stock with a current price S=100, a volatility σ=10%, and a riskless rate r=5% (riskless rates are the government bonds interest rates). Line 101 (“C”) shows the price of the call option with respect to stock price S (shown as the X axis). The price of the option 101 forms a curve and a measure of the curvature (i.e. non-linearity) of the line at any particular point is its gamma. Line 104 shows the value of an initial quantity of stock (Δ) that was sold short by the trader to hedge the option position. The value of the quantity of stock (Δ) sold short 104 is shown as decreasing linearly when the option price is increasing, and increasing linearly when the option price is decreasing. This linear behavior of (Δ) provides a relatively good hedge for a small movement of the stock price.
Line 103 shows the combined value of the long option position 101 and the short Δ stock 104 when there is no rebalancing of the short position Δ 104. Due to the curvature of the option price given by the gamma, this combination will produce a profit (line 102). This profit 102 is usually called long gamma trading profit. If, when the option matures, this profit is greater than the premium paid to buy the call option, then the trader will have made a gain, otherwise there will be a loss. A gamma trading strategy tries to maximize the profit or minimize the loss generated by the delta hedging of an option position.
For a gamma trading strategy, the magnitude and frequency of the swings of the stock price (which can be measured in relation to an initial reference price such as the initial stock price “S” or the initial strike price “K”) typically has a more significant effect on profit and loss than does the direction of the stock movement. This magnitude and frequency is measured by the realized volatility of the stock during the lifetime of the option, (σr). If the realized volatility (σr) is greater than the volatility implied by the option price when the option was bought (σ), then the strategy will have a profitable outcome. It is noted that implied volatility (σ) may be calculated using the well-known Black-Scholes model.
If a trader is primarily interested in closely hedging several option positions each with a long gamma, the trader has to frequently monitor the price of the several underlying stocks and rebalance the hedge position for each stock by buying or selling an amount of stock (Δ0−Δ1) in response to changes in each stock price. Manual re-balancing of a hedge position in this manner can be time consuming as it requires the option trader to closely monitor the prices of all the stocks in the trader's portfolio. Furthermore, to ensure that desired hedging trades are executed, a trader may resort to buying at the best offer price (rather than placing a limit order at the lower best bid price), or selling at the best bid price (rather than placing a limit order at a higher best offer price). This buying at the best offer price and selling at the best bid price, referred to as paying the bid-ask spread, may reduce the trader's profitability.
Delta hedging and gamma trading are explained in greater detail in, e.g., Baz Jamil, Vasant Naik, David Prieul, Vlad Putyatin, Francis Yared, Selling risk at a premium, Risk, December 2000, pages 135-138; Hull John, Introduction to Futures and Options Markets, second edition, Prentice Hall International Edition, 1995, pages 319-345.